The Behajaina method applied to the bodies of fractions twisted by a derivation
The Inverse Galois Problem (IGP) over a commutative field \(k\) asks whether every finite group is the Galois group of a Galois extension of \(k\). Going back to Hilbert and Noether, the original problem over \(\mathbb{Q}\) remains widely open. Traditionally, this problem has been investigated only over commutative fields. However, there is a notion of Galois extension for arbitrary extensions of skew fields, which is due to Artin: an extension of skew fields \(L/K\) is \textit{Galois} if \(K\) is the invariant subfield of \(L\) under the natural action of \(\Aut(L/K)\). Moreover, when the degree of \(L/K\) is finite, there is also a Galois correspondence generalizing the classical one [\textit{P. M. Cohn}, Skew fields. Theory of general division rings. Cambridge: Cambridge Univ. Press (1995; Zbl 0840.16001)]. Hence, it makes sense to study the IGP over any skew field. \textit{B. Deschamps} and \textit{F. Legrand} [J. Pure Appl. Algebra 224, No. 5, Article ID 106240, 13 p. (2020; Zbl 1480.12004)] started the investigation of this problem and proved that: if \(H\) is a skew field of finite dimension over its center \(k\), which contains a so called ``ample field'', then the IGP has a positive answer over the skew field of rational fractions \(H(t)\). The proof of their result is based on scalar extensions method by using reduced norms. \textit{A. Behajaina} [J. Pure Appl. Algebra 225, No. 4, Article ID 106549, 10 p. (2021; Zbl 1473.12002)] generalized this result to skew fields of rational fractions \(H(t,\sigma)\) (here \(th=\sigma(h)t\) for all \(h \in H\)), where \(\sigma\) is an automorphism of \(H\) of finite order such that the invariant subfield \(k^{\sigma}\) of \(k\) contains an ample field. In this result, the condition that \(H\) is of finite dimension can be relaxed. The key ingredient of the proof is a certain ''embedding'' into the skew field of series \(H((t,\sigma))\), which replaces the use of ''reduced norms'' by Deschamps and Legrand, which heavily depends on the finiteness of \(H\) over \(k\). In this paper, the author proves, following the method of Behajaina, that: if \(H\) is a skew field of center \(k\) of positive characteristic, \(\sigma\) is an automorphism of \(H\) and \(\delta\) is a \(\sigma\)-derivation (that is, \(\delta\) is an additive morphism satisfying \(\delta(xy)=\delta(x)y+\sigma(x)\delta(y)\), for all \(x,y \in H\)) such that \begin{itemize} \item \[ k^{\sigma,\delta}=\{ x \in H \mid \sigma(x)=x, \delta(x)=0\} \] contains an ample field, \item and \(t^n\) is in the center of \(H(t,\sigma,\delta)\) for some positive integer \(n \geq 1\), \end{itemize} then the IGP has a positive over the skew field of rational fractions \(H(t,\sigma,\delta)\) (here \(th=\sigma(h)t+\delta(h)\) for all \(h \in H\)). This result generalizes the result of Behajaina by taking \(\delta=0\) and \(n\) the order of \(\sigma\). Also, the author provides some applications which cannot be deduced from the result of Behajaina.
scientific article; zbMATH DE number 7357694
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